Download Page 1 Lecture: Quantum Optics Derivation of the Master Equation

Lecture: Quantum Optics
Derivation of the Master Equation (System/Reservoir Interactions)
General Description of "Open" Quantum Systems
In our discussion of the Optical Bloch Equations we included phenomenological decay
constants (1/T1 and 1/T2). We accounted for these constants in terms of spontaneous
emission, which arises from the coupling of the atom to the vacuum modes of the quantized
electromagnetic field. The excitation is irreversibly emitted from the atom into the field,
never again to be reabsorbed in a Rabi flop, in contrast to the interaction of an atom with a
coherent light source. The evolution of an excited atom "immersed" in the vacuum of the
electromagnetic field is a particular example of an "open" quantum system. That is, if all we
are interested in is the evolution of the atom, we cannot describe it by a closed system with a
finite number of degrees of freedom. Instead, the atom is coupled to the "outside" world (in
this case the modes of the vacuum). Generally the outside world will consist of a huge
number of degrees of freedom, and therefore we have no hope of following all of them. If
fact we might not even know the exact state of the outside world, having only some
statistical information to describe it.
One generally describes such a situation as the interaction of a "system" with a
"reservoir". The system contains the (few) degrees of freedom we want to follow in detail
(for example the internal state of our two level atom), while the reservoir contains the
multitude of degrees of freedom associated with the outside world (for example the modes
of the electromagnetic field).
Reservoir
R
System
S
HR
Hs
We assume that there is some Hamiltonian HS which describes the evolution of system
alone, a Hamiltonian for the reservoir HR and a Hamiltonian that describes their weak
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interaction, HSR, so that the Hamiltonian for the entire (closed) system+reservoir Hilbert
space is
Htotal = HS + HR + HS R.
As the reservoir is supposed to be large it will relax on a very short time scale, and hardly
"notices" that the system is interacting with it. On the other hand, the small system is
profoundly effected by its coupling to the reservoir. Any excitation of the system which
gets coupled to the reservoir is effectively lost forever. In this way, energy is (effectively)
irreversibly dissipated into the reservoir, and the entropy of the system increases. This
description is familiar in classical statistical mechanics. For example, thermal equilibrium is
established when a small system in placed in contact with a "bath" or "reservoir" at some
temperature T. There too, one assumes that the reservoir is barely effected by the system,
relaxing back to its equilibrium in a infinitesimally short time. Of course this is an
approximation. It remains one of the great unsolved problems of theoretical physics to
reconcile the second law of thermodynamics, with its intrinsic arrow of time, with the
microscopic time-reversible dynamical equations. The problem of dissipation in quantum
mechanics has all the same ingredients. The new ingredient is that the reservoir may contain
not only thermal fluctuations, but also quantum fluctuations associated, so even at absolute
zero, the reservoir can cause decay of the excitations of the system.
Because of the dissipative nature of this problem we must turn to the density matrix for
a proper description. Suppose at some initial time t the system is placed in contact with the
reservoir. Up until that point S and R evolved freely with no coupling, so the total density
matrix can be written simply as a product
ρ total(t) = ρ s (t) ⊗ ρ R (t).
For any nontrivial interaction, excitations of the system will get "entangled" with the
excitations of the reservoir, (see lecture #3 for the definition of entangled states),
ρ total(t + ∆t) ≠ ρ s (t + ∆t) ⊗ ρ R (t + ∆t) .
Since we are only interested in the system degrees of freedom, without any regard to the
state of the reservoir, we trace over the degrees of freedom of the reservoir. The result is a
reduced density operator for the system alone,
ρ s (t + ∆t) = TrR (ρtotal(t + ∆t )) .
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Because of the entanglement, even if the initial state of the system were a pure state, at this
later time, the reduced density matrix will be a mixed state. Thus, the entropy of the system
has increased.
Our goal in this lecture is to derive the equation of motion for the reduced density
matrix for the system. We start with what we know: the closed system is Hamiltonian, and
thus evolves according to the Schrödinger equation (written here in terms of the density
operator)
i
i
i
ρ˙ total = − [Htotal, ρ total] = − [HS + HR ,ρ total] − [HS R, ρ total].
h
h
h
The equation of motion for the reduced density matrix of the system then follows by taking
the trace of this equation over the reservoir variables
ρ˙ s = TrR (ρ˙total) .
There are two key ingredients which will go in to our derivation
• The reservoir is large and relatively unaffected by its interaction with the system.
• Any "memory" of an effect S has on R is lost in a time scale much shorter than the time
scale that S relaxes. (Stated alternatively, the reservoir is very broadband in frequency).
For concreteness we will specialized to the problem of a two-level atom in contact with a
"bath" of electromagnetic blackbody radiation at temperature T. The problem of
spontaneous emission in the quantum vacuum is just the particular case T=0.
Two Level Atom Interacting with Thermal Radiation
To make contact with the discussion above we make the following associations:
• System = two level atom
• Reservoir = modes of the electromagnetic field
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The total Hamiltonian for S+R is Htotal = H A + HF + Hint, where
HA = hω eg e e = hω eg S+ S−
1

HF = ∑ hω k ak† ,λ ak,λ + 

2
k ,λ
Hint = −d ⋅E ≈ − e d ⋅ E
(+)
g S+ − e d ⋅ E
( −)
g S− .
In the expressions above we have used the usual equivalence between our two level atom
and a spin 1/2, S+ = e g , S− = g e , and in the interaction Hamiltonian we used the
rotating wave approximation, with positive and negative frequency components of the field
given in lecture #7. If we substitute the usual normal mode expansion for the
electromagnetic field,
E
(+)
(x) = −i ∑
k, λ
2πhω k r
ε k, λ ak, λ e ik⋅ x ,
V
and take our atom at the origin, then we can express the interaction Hamiltonian as
(
†
)
Hint = h FS+ + F S− , where F = −i ∑
k, λ
r
V k,λ
2 πhω k e ε k,λ ⋅ d g
ak, λ = ∑
ak, λ
V
h
k, λ h
2 πhω k r
e ε k,λ ⋅ d g . Note, V k, λ is
V
nothing more that the dipole interaction energy of the atom with a single photon.
hΩ k, λ
Sometimes we will write V k, λ =
, where Ω k, λ is the single photon Rabi frequency
2
(think about the factor of 1/2 here).
At the initial time t the atom and field are uncorrelated
Here we've defined coupling constants, V k, λ = −i
ρ total(t) = ρ A (t) ⊗ ρ R (t) ,
The initial state of the atom ρ A (t) is arbitrary (maybe it's a pure mixture of |e> and |g>), and
the state of the field is to describe a thermal distribution of photons (blackbody radiation) at
some temperature T,
ρ R (t) =
 − HR 
1
 = ∑ P(nk ,λ ) nk, λ nk, λ ,
exp
Z
 k B T  k, λ
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  − HR  
  = Partition function ,
Z = Trexp

k
T

B 
−hω /k T
e k B
P(nk, λ ) =
= Probability of nk, λ photons in the mode (k, λ ).
1− e hω k /k BT
In other words, the state of the field is described by a canonical ensemble, which in our
quantum mechanical language, is a statistical mixture of number-states, weighted by the
Boltzman probability factor, normalized by the partition function. The average values of
observables are calculated exactly as in classical statistical mechanics. For example, the
average number of photons in a mode is,
n (ω k ) = Tr(ρR a†k,λ ak, λ ) = ∑ P(nk, λ ) nk,λ ak† ,λ ak,λ nk, λ
k,λ
= ∑ P(nk, λ )nk, λ =
k,λ
1
hω k /k BT
e
−1
(Planck Distribution )
.
Similarly we find,
†
†
Tr(ρR ak, λ ) = Tr(ρ R ak† ,λ ) = Tr(ρR ak, λ ak ′, λ ′ ) = Tr(ρR ak,
λ ak ′, λ ′ ) = 0.
That is to say, only operators which are diagonal in the number-state basis have a nonzero
expectation value. Physically, broadband thermal radiation has a completely random phase.
Thus, an operator which depends on the phase has no average value in a thermal state.
Equation of motion for the reduced matrix
In order to remove the trivial free evolution of the atom and field from our problem will
we transform into the "interaction picture" via the unitary operator,
i
U( I ) = exp− ( HA + HF )t  . The density operator, and interaction operator in this picture
 h

are
†
(I)
ρ total
(t) = U( I ) ρ totalU( I )
†
(
†
)
(I)
Hint
(t) = U ( I ) HintU ( I) = h F ( I ) (t)S(+I ) (t) + F( I ) (t)S−( I )(t) ,
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(I )
with F (t) = ∑
k, λ
V k, λ
h
ak, λ e
− iω k t
− iω egt
(I)
, S+ (t) = S+ e
. Using these definitions, and the
Liouville equation in the Schrödinger picture, we arrive at the Liouville equation in the
interaction picture,
i (I)
(I)
(I)
ρ˙ total
= − [Hint (t), ρ total(t)]
h
If we integrate this equation from t→t+∆t, we obtain the formal solution,
(I)
(I)
ρ total
(t + ∆t) = ρ (t) −
i t +∆t
( I)
(I)
∫ dt ′ [Hint ( t ′), ρtotal(t ′)] .
h t
This can be iterated over and over to get an infinite series solution. If we terminate this
series at some point we get a perturbation solution, good to some order in Hint. Since the
coupling between the system and reservoir is assumed to be weak will expand only up to
second order
(I)
(I)
ρ total
(t + ∆t) = ρ total(t) −
i t+ ∆t
(I)
(I)
∫ dt ′ [Hint (t ′ ), ρ total(t)]
h t
1 t+ ∆t t ′
( I)
I)
I)
− 2 ∫ dt ′ ∫ dt ′′ [Hint
( t ′), [H(int
(t ′′), ρ (total
(t)]
h t
t
.
Henceforth we will drop the superscript (I), with the understanding that we are operating in
the interaction picture.
We seek the dynamics of the reduced density matrix for the atom alone, having traced
over the modes of the electromagnetic field,
(
ρ A (t + ∆t) = TrR ρ total(t + ∆t)
= ρ A (t) −
)
(
)
i t +∆t
∫ dt ′ TrR [Hint(t ′), ρ A (t )ρ R (t)]
h t
−
(
)
.
1 t+ ∆t t ′
∫ dt ′ ∫ dt ′′ TrR [Hint( t ′), [Hint(t ′′ ), ρ A (t)ρR (t)]
h2 t
t
Here we have used the fact that at the initial time the atom and field are uncorrelated so that
the total density matrix is a product of the two subsystems. Substituting for Hint(t) given
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on page 5, we arrive at the following expression, after some tedious algebra, making use of
the cyclic property of the trace Tr(ABC)=Tr(CAB)=Tr(BCA),
+ [S ( t ′), ρ (t)] F (t ′) )
(
dt ′ ∫ dt ′′ {( S ( t ′)S ( t ′′)ρ (t) − S (t ′ ′)ρ (t)S (t ′)) F (t ′)F(t ′′)
t+ ∆t
ρ A (t + ∆t) − ρ A (t) = − i ∫ dt ′ [S+ (t ′ ), ρ A (t)] F(t ′)
t + ∆t
– ∫
t
(
t
t′
−
t
+
+
A
)
+ S+ ( t ′)S− (t ′′)ρA (t) − S− (t ′′ )ρA (t)S+ (t ′ ) F(t ′)F † (t ′ ′)
−
R
†
A
A
R
†
−
R
+ H.c.}
†
†
+ Terms proportional to F(ta )F(tb ) or F (ta )F (tb ) .
(Whew!!).
Here, H.c. stands for the Hermitian conjugate to the preceding terms, and
R
is a short
hand for the expectation value of some field operator take in the state ρR(t) given on page 4,
O R ≡ Tr(OρR ). Using the definition of F(t), and the fact that the average value of any
field operator which depends on the phase of the field vanishes we have,
F( t ′)
R ≡ Tr(F(t ′)ρ R (t)) = ∑
k ,λ
†
Similarly, we find F ( t ′)
R
Vk, λ
2
= F(ta )F(tb )
R
e − iω k t ′ Tr(ρ R (t )ak,λ ) = 0 .
†
†
= F (t a )F (tb )
with the second order terms proportional to F † (ta )F(t b )
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R
R
= 0 . Thus we are left only
or F(ta )F† (tb )
R
.
R
Reservoir Correlation Function and the Markov Approximation
Let us define
†
K(t ′, t ′′ ) ≡ F (t ′)F(t ′′)
R
,
known as the reservoir correlation function. Physically this measures how an excitation
of the reservoir at time t' is correlated with the excitation at t". From general principles, we
can immediately see some properties of K.
• Stationary correlations
For a reservoir describing thermal equilibrium the correlations should not depend on any
origin of time. That is if we ask how an excitation at t' is correlated with an excitation at
t"=t'+τ, the result should only depend on the time difference τ,
K(t ′, t ′′ ) = K( t ′ − t ′ ′) = F† (t ′ − t ′′ )F(0)
R
• Time reversal
From the properties of stationarity and the Hermitian conjugate we have
K(t ′′ − t ′) = F † (t ′ ′)F( t ′) = K* (t ′ − t ′′ ).
R
Thus, the real part of K is symmetric in its argument: Re(K(−τ )) = Re(K( τ )).
From these properties we can make a rough sketch of the of the correlation function
Κ(τ)
τc
τ
For sufficiently large values of |τ|, the correlation function should approach zero. The
characteristic temporal width τc is known as the correlation time. Physically, it sets the
scale of time over which the reservoir relaxes back to thermal equilibrium given some
excitation. Figuratively, one often speaks of the correlation time as the time scale over
which the reservoir has "memory". After this time the reservoir is back to equilibrium and
effectively has no memory that there was any excitation.
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One can also define a spectral density of the correlation function by as the Fourier
transform
iωτ
S(ω ) ≡ ∫ dτ K( τ )e
.
The bandwidth of the reservoir, ∆ωR, given by the characteristic width of S(ω), is related to
the correlation time by the usual Fourier duality
τc ~
1
.
∆ω R
As ∆ ω R → ∞ , (i.e. the reservoir is "white noise"), the correlation time τ c → 0. The real
part of the correlation function is then proportional to a delta function
Re(K( τ )) ~ δ (τ ).
Thus a "white-noise" reservoir has no memory; it instantaneously relaxes to its equilibrium
value. In general, the correlation time will be very small, but not exactly zero.
The other important time scale in the problem is the time for the system to relax to its
equilibrium value as it dissipates energy to the reservoir. If we call this rate of relation Γ,
then it will generally be the case that Γ<<∆ωR, or equivalently τc<<1/Γ. Furthermore, this
inequality is generally so extreme that the reservoir correlation time is effectively zero when
compared with the system decay time. We want to study the evolution of the reduced
density matrix for the system alone, which changes on a time scale 1/Γ. Thus, we want to
look on time scales ∆t<<1/Γ. On the other hand, this time scale may still be very large
compared to the reservoir correlation time. Therefore, we have
τ c << ∆t <<1 / Γ .
Stated alternatively, we want to study the dynamics of the reduced density matrix for the
system "coarse-grained" over the reservoir correlation time. In this way we approximate the
reservoir as have no memory, and is thus "delta-correlated". Such an approximation is
known as the Markov approximation. This approximation will allow us to take the timereversible equations of motion into irreversible equation which give rise to dissipation.
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Master Equation in the Markov Approximation
With our characterization of the correlation function for the reservoir forcing function,
we make the following change of variables so that, only time differences appear,
τ ≡ t ′′ − t ′, ξ ≡ t ′′ − t .
The Jacobian for the change of area element for out two dimensional integral is
 ∂ξ
∂(ξ , τ )
= det  ∂∂tτ′
∂(t ′, t ′′)

 ∂t ′
∂ξ 
∂t ′′  = det  0 1 = 1.
 −1 1
∂τ 
∂t ′′ 
To get the limits of integration we must may the two dimension integration region in the
{t',t") plane to the (τ,ξ) plane
t'
B
t+∆t
A
t
B
C
t
Thus,
ξ
t'=t"
∆t
t+∆t
t + ∆t t ′
A
t"
∆t
∆t
0
0
∆t
C
τ
∫ dt ′ ∫ dt ′′ → ∫ dξ ∫ dτ . The equation for the reduced density operator for the atom
t
t
in these new variables becomes
∆t
∆t
0
0
ρ A (t + ∆t) − ρ A (t) = − ∫ dξ ∫ dτ
{( S S ρ (t) − S ρ (t)S )e
− + A
(
+ A
−
iω egτ
)
+ S+ S− ρ A (t) − S−ρ A (t)S+ e
+ H.c.}.
F† (0)F(τ )
−iω egτ
F(0)F † ( τ )
Here we have substituted for S+ and S– in terms of their simple exponential time
dependence.
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R
R
Now we invoke the Markov approximation. According to the discussion above, the
integrand will be nonnegligible only for times τ smaller than the correlation time. As we are
coarse graining the dynamics on time scales ∆t>>tc, the upper limit of integration over τ is
effectively infinity, (∆t→∞, τ integral). On the other hand, the coarse grained time step is
chosen to be much much smaller than the rate at with ρA changes, as is therefore, effectively
a differential. Under the Markov approximation we have
(
)
∞
dρ A
iω τ
†
≈ −S−S+ρ A (t ) + S+ρ A (t)S− ∫ dτ e eg F (0)F(τ )
dt
0
(
)0
∞
+ −S+S−ρ A (t) + S−ρ A (t)S+ ∫ dτ e
+ H.c.
−iω egτ
F(0)F ( τ )
†
R
R
The Fluctuation-Dissipation Theorem
We are left to evaluate the Fourier transforms of the correlation functions. Substituting
for the reservoir forcing function F(t) on page 4, gives.
∞
∫ dτ e
iω egτ
0
F (0)F( τ )
†
R
We found on page 5, a†k, λ ak ′, λ ′
∞
∫ dτ e
iω egτ
0
F (0)F( τ )
†
R
=
R
∑
k,λ ,k ′, λ ′
V *k,λ Vk ′, λ ′
h
†
ak, λ ak ′, λ ′
2
∞
R
∫ dτ e
−i( ω k − ω eg )τ
0
.
= n (ω k ) δ k,k ′ δλ ,λ ′ , so that
= ∑
k,λ
V k, λ
2
h
2
∞
n (ω k ) ∫ dτ e
−i(ω k −ω eg )τ
0
.
The next step in obtaining irreversible equations of motion is to let the reservoir consist of an
uncountable infinite (i.e. continuous) number of degrees of freedom. Under this approximation
the sum over reservoir modes goes to an integral over the density of modes D(ω k ),
∞
∫ dτ e
0
iω egτ
F (0)F( τ )
†
R
= ∑ ∫ dω k
λ
V λ (ω k )
h
2
2
∞
D(ω k )n (ω k ) ∫ dτ e
− i( ω k −ω eg ) τ
0
Finally, it remains to evaluate the integral over τ. In order to do this we add convergence
factor e–ετ, and then take the limit as ε→0,
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.
∞
lim ∫ dτ e
ε→0 0
−i(ω k − ω eg −iε )τ
1
ε →0 i(ω k − ω eg) + ε
ω k − ω eg
ε
= lim
2
2 − i lim
2
2.
ε → 0 (ω k − ω eg ) + ε
ε →0 (ω k − ω eg ) + ε
= lim
The real part is a Lorentzian, which in the limit ε→0, becomes a delta function whose area is
π,
ε
2
2 = π δ (ω k − ω eg) .
ε → 0 ( ω k − ω eg ) + ε
lim
In this limit, the imaginary part become Cauchy's "principle part",


1

,
=
P
2
2
ε → 0 ( ω k − ω eg ) + ε
 ω k − ω eg 
lim
ω k − ω eg
defined, like the delta function, by its effect integration
∞  f (x)
 −ε
1
∫ dxP   f (x ) ≡ lim  ∫ dx + ∫ dx 
.
x
ε →0  −∞
 x
ε
In other words, the principle part allows one to exclude the singularity at x=0. Thus,
∞
∫ dτ e
− i(ω k −ω eg ) τ
0


1
.
= π δ (ω k − ω eg ) − iP 
 ω k − ω eg 
Plugging this into the equation of motion for the reduced density matrix
∞
∫ dτ e
0
iω egτ
F † (0)F( τ )
R
=
2
1 2π
∑ 2 Vλ ( ω eg) D(ω eg )n ( ω eg)
2λ h
2
V (ω k )
i
− P ∫ dω k λ
D(ω k )n (ω k ).
h
hω k − hω eg
We recognize the real part of this expression as the transition rate for the perturbation
Hamiltonian as given by Fermi's Golden rule. In fact, if we separate off the factor n (ω eg) ,
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then the remaining coefficient is nothing more than the spontaneous emission coefficient
(the Einstein A-coefficient) calculated in lecture #8!
Γ=∑
λ
2
2π
2 V λ (ω eg ) D(ω eg ) .
h
The imaginary part consists of a square of the coupling strength dived by an energy
denominator. This is recognizable as the correction to the energy levels of the free atom due
to the light field as given by second order perturbation theory,
δ Elight = P ∫ dω k
Vλ (ω k )
2
hω k − hω eg
D(ω k )n (ω k ).
This energy shift is known as the "light-shift" and will typically be a small fraction of the
energy separation between the ground and excited states. Putting this all together,
∞
∫ dτ e
iω egτ
0
F † (0)F( τ )
R
=
1
i
n Γ − δEight,
2
h
where n is the Planck distribution evaluated at the resonance frequency of the atom and
temperature of the reservoir,
n ≡ n (ω eg ) =
1
.
exp(hω eg / k BT ) − 1
The calculation for the Fourier transform of the correlation function F(0)F † ( τ )
R
follows in exactly the same manner. The only difference is that when we substitute for the
reservoir function in terms of the normal mode expansion we have the reservoir average,
ak, λ a†k ′, λ ′
R
= (n (ω k ) + 1) δ k,k ′ δ λ , λ ′ .
The factor of 1 comes from the commutator of ak, λ and a†k′ , λ ′ , and thus represents the
effects of vacuum fluctuations. After carrying through the calculation one finds,
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∞
∫ dτ e
0
− iω eg τ
F(0)F (τ )
where δ Evac = P ∫ dω k
†
R
Vλ ( ω k )
=
2
hω k − hω eg
1
i
i
(n +1)Γ + δ Elight + δEvac ,
2
h
h
D(ω k ) is the energy shift due to the vacuum. This is
one of the contributions to the famous Lamb shift of quantum electrodynamics. We do not
get the full Lamb shift in this calculation as we have treated the atom in a nonrelativistically.
If we look at the real part of the correlation functions is time, we have
1
Γn δ (τ ),
2
1
Re F(0)F † ( τ ) = Γ(n + 1) δ ( τ ) .
R
2
Re F † (0)F( τ )
R
=
These have the characteristic "delta-correlations" associated with the Markov approximation
as discussed above. In addition we see that the "strength" of the correlations, given by the
area under the correlation function is proportional to the decay constant Γ. This is an
important result, originally derived by Einstein in the context of classical Brownian motion,
which relates the strength of the fluctuations of the reservoir to the damping constant of the
system. This is known as the fluctuation-dissipation theorem.
The Master Equation for thermal radiation
Having determined the Fourier transform of the correlation functions we can write the
equation of motion for the reduced density matrix. Recall that were operating in the
interaction picture. We return to the Schrödinger picture at this state by restoring the free
atom evolution. Plugging in for the correlation functions, and including the H.c. terms,
dρ A
i
Γ
= − [ H′A , ρ A ] − n{S− S+ , ρ A} + Γ n S+ρ A S−
dt
h
2
Γ
− (n +1){S+ S− , ρ A} + Γ (n + 1)S−ρ A S+ .
2
Here { , } is the anticommutator, and
H′A = (hω eg − δ Elight)S+ S− + (δElight + δEvac )S− S+
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= (hω eg − δ Elight) e e + (δ Elight + δEvac ) g g
is the renormalized atomic Hamiltonian, including the energy level shifts due to the lightshift and the vacuum shift. Using the definition of the S+ and S– we can write the equations
of motion for the matrix elements of ρA,
ρ˙ ee = e ρ A e = −Γ(n +1)ρee + Γn ρ g g
ρ˙ g g = g ρ A g = −Γn ρ g g + Γ(n + 1)ρee
Γ
ρ˙ eg = e ρ A g = −i(ω ′eg − i (n +1)) ρeg
2
Note that ρ˙ ee + ρ˙ g g = 0, so the trace of the density matrix is preserved. The first two
equation are nothing more that Einstein's rate equation, with the contributions of stimulated
absorption and emission, as well as spontaneous emission evident. The third equation tells
us how the coherence in the superposition of excited and ground states decays. The steady
state solution is immediately found by setting the right hand side to zero,
ρ eg = 0
−1



ρee
n
1
1

=
=
+ 1 = exp(−hω eg / kB T )
ρ g g n +1  exp(hω eg / k B T ) − 1  exp(hω eg / k B T ) − 1 
The coherences decay to zero, while the populations reach a steady state distribution which
is a thermal Boltzman distribution
ρee e − Ee / kB T
− (E − E ) /k T
−hω / k T
= − E g / kB T = e e g B = e eg B .
ρg g e
Thus, we come to the physically appealing solution that, in steady state, the atom comes into
thermal equilibrium with the "bath" of thermal radiation, and can be written as a canonical
ensemble at the same temperature T as the reservoir.
Steady State: ρ A =
e − H A / kB T
.
Z
We have come full circle to the solution derived by Einstein. He assumed a thermal
distribution and rate equations, and was able to derive a relation between the spontaneous
Page 15
and stimulated emission coefficients. We assume only fundamental interactions of
absorption and emission, and were able to derive the coefficients, and the steady state
thermal distribution which results.
The connection between system-reservoir interactions and quantum measurement
We see that even if the initial state of the atom were a pure state coherent superposition
of excited and ground states, in steady state it will "decohere", i.e. decay, into a statistically
mixed state with no phase coherence. Phase coherence was lost in the |e> |g> basis
because of the form of the S-R coupling and the fact that the state reservoir had a random
phase. The loss of phase coherence due the interaction with the reservoir is the central
feature of this problem. In general a measuring device, in Bohr's sense of a macroscopic
classical apparatus, can be though of as a reservoir, which will thus act to decohere a pure
state into a mixed state. The mixed state is a classical ensemble of possibilities, "e" or "g"
as opposed to the quantum superposition where the atom has the "potential" for being
measured "e" or "g". This decoherence can be thought of as the mystical "collapse" of the
wave function, which projects a wave function onto an eigenstate of a the observable being
measured. Here the collapse is seen as a decoherence from a pure state to a mixed state.
We can never, within the confines of the quantum mechanical formalism, describe the
evolution of an individual member of the ensemble. Thus this result must only be
interpreted in a statistical sense of the evolution of an ensemble.
The Master Equation for an Atom in the Vacuum, with a Coherent Driving Field
The case of the atomic damping due to the electromagnetic vacuum is already contained
in our solution. It is the particular case of a zero temperature reservoir with zero average
photon number n = 0 . In the absence of any other field, the Master equation becomes
dρ A
i
Γ
= − [ H′A , ρ A ] − {S+S− , ρ A } + Γ S−ρ A S+
dt
h
2
Γ
= − iω eg
′ [S+ S− , ρ A ] − {S+S− , ρ A } + Γ S−ρ AS+ .
2
The evolution of the matrix elements can be written immediately,
ρ˙ ee = − ρ˙g g = −Γρee
Γ
*
ρ˙ eg = ρ˙ge
= −i(ω ′eg − )ρ eg .
2
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Thus we have justified our phenomenological decay constants 1/T1 and 1/T2 from lecture5.
Notice that the anticommutator term in the Master equation has the same form as the
atomic Hamiltonian term. It is sometimes useful to define an effective nonHermitian
atomic Hamiltonian
Γ
Γ


Heff
′ − i  S+ S− = hω eg
′ −i  e e .
A ≡  hω eg

2
2
The master equation can then be written,
dρ A
i eff
= − [HA ,ρ A ]′ + Γ S−ρ A S+ ,
dt
h
eff
eff
eff †
where [H A , ρ A ]′ ≡ HA ρ A − ρ A HA
, so that ρA is Hermitian. Thus, the anticommutator
term accounts for decay of the excited state, while the second is a "feeding" term which
accounts for the replenishing of the ground state when the atom makes a "quantum jump"
from the excited state into the ground state. This division of the relaxation effects into
decay and quantum jumps is useful for numerical solutions to the master equation. One can
simulate the "trajectory" of the atom initially in the excited state by a deterministic decay by
the nonHermitian Hamiltonian, followed by a quantum jump. Averaging over many
realization of this random event givens the solution. This method is known as the
quantum Monte-Carlo wave function simulation and very useful for extremely large
problems were there are too many matrix elements to solve for using more traditional
techniques.
If we want to study the evolution of the atom in the presence of a classical, coherent,
driving field, such as a monochromatic laser, but also including spontaneous emission, we
need only include the dipole Hamiltonian for the atom-field interaction Hint,
dρ A
i
i
Γ
= − [ H′A , ρ A ]− [Hint, ρ A ]− {S+ S− ,ρ A} + Γ S−ρ A S+ .
dt
h
h
2
Then in the rotating frame, and using the rotating-wave approximation, we arrive at the
equations for the evolution of the matrix elements,
Ω
ρ˙ ee = − ρ˙g g = −Γρee + i (ρ˜ ge − ρ˜ eg )
2
Γ
Ω
*
ρ˙˜ ge = ρ˙˜eg
= −i(∆ − i )ρeg + i (ρee − ρ g g ).
2
2
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This is the standard master equation for the two level atom that we have been studying for
many weeks.
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